A New Method For Designing Three-dimensional Coils In Advanced Stellarators

Stellarat.ors are magnetically confined fusion experiments that use non-axisymmetric magnetic field produced by currents in external coils to confine plasma.Toroidal plasma currents are not necessary for stellarators.Therefore.stellarators are inherently able to avoid current drive instabilities and operate in steady state.For this reason,they become increasingly focused.The complexity of three-dimensional coils is still a critical challenge.In order to find appropriate coils pro-ducing desired magnetic field,conventional approaches assume a toroidal "winding suface",on which the coils lie,and obtain coil geometries by approximating with a surface current potential contours or by nonlinear optimization techniques.The presence of winding surface r estricts coils from moving in space freely,and to some degree it makes the optimization procedure more complicated.In this thesis,we will introduce a new method for designing stellarator coils.Coils are described as space curves and represented by a Fourier representation.The parameters of coils are then adjusted by nonlinear optimization algorithms to minimize a predefined target function.The target function comprises multi-ple cost function that includes both physical requirements and engineering con-straints,e.g.the boundary condition of satisfying the normal field.The first and second order derivatives of the target function are calculated by analytical expressions to compute the gradient and Hessian fast and accurately,such that nonlinear optimization algorithms,like the famous Newton method.are applied.A numerical code,named Flexible Optimized Coils Using Space curves(FOCUS),has been developed.FOCUS was verified and validated by an analytical model and a simple stellarator configuration.Afterwards,FOCUS was used to design and optimize different types of coils in several stellarators.including reproducing the modular coils of W7-X.improving HSX coils,corrcting error fields of LHD helical coils.Besides,FOCUS was applied to design resonant magnetic perturba-tion(RMP)coils in DIII-D tokamak.Unlike existing approaches,the new idea to design RMP coils starts with the desired plasma properties to be controlled.Magnetohydrodynamics(MHD)perturbed equilibrium code GPEC was used to calculate the required magnetic perturbation,followed with using FOCUS to find coils that can produce those perturbations.We also present a new method for determining the error field sensitivity to coil deviations by using the eigenvalues and eigenvectors of the Hessian in FOCUS.In this method,the error field evaluation function was expanded by a Taylor series in the neighborhood of a local minimum.The relationships between the evaluation function and the eigenvalues of Hessian were derived.It was validated on a CNT-like configuration by comparing existing approaches.In summary.coil optimizations are simplified by getting rid of the winding surface with coils represented as space curves.Benefiting from analytical calcula-tions of the gradient and Hessian of the target function,the new coil design code,FOCUS,has been proven to becme much faster and more robust than previous approaches.Consecutive method for designing RMP coils provides new ideas.Determining error field sensitivity to coil deviations appears elegant and efficient.